On global solutions for non-linear Hamiltonian evolution equations
Marsden, J. E.
Comm. Math. Phys., 30, (1973), 79-81
It is shown that partial differential equations of Hamiltonian type admit global solutions in time if (a) the initial data is near equilibrium (or the coupling constant is small) (b) the linear terms have positive energy and (c) the non-linear terms are smooth functions in the topology of the linearized energy norm. The non-linear terms need not have positive energy. The result is applied to non-linear wave equations in which the interaction energy is not necessarily positive.